Trump lawyer Rudy Giuliani had expressed a similar sentiment when he told the Times that while there was a discussion, the payment was never made and the tape showed that Trump had done nothing wrong.
"Inconceivable that the government would break into a lawyer's office (early in the morning) - nearly unheard of", the president tweeted.
He then said "even more inconceivable... that a lawyer would tape a client... totally unheard of and perhaps illegal".
Guardiola says Manchester City trying to find solution for 'professional' Joe Hart
Guardiola allowed defensive midfielder Yaya Toure to leave in the summer and wants a replacement. The vast majority of his first team will join up when they are back in Manchester.
Trump insisted on his innocence in Saturday's tweet, assuring his followers that "the good news is that your favorite President did nothing wrong!".
Michael Cohen recorded the conversation with Trump, which was seized by federal agents.
The prosecutors working on the case had not reviewed the recording because it, along with millions of other documents and files seized in Federal Bureau of Investigation searches of Cohen's home, hotel room and office in April, was undergoing the special master process, in which an independent party reviews whether the items should be regarded as privileged and thus withheld from prosecutors.
100 ex-students report sex misconduct by former Ohio State University doctor
Some accusers have also claimed former OSU athletics staff - including US Rep. Jim Jordan, a former OSU assistant wrestling coach - knew about the abuse.
In May, President Trump admitted that he had reimbursed Mr Cohen for a payment to another woman to hush up her claims of an affair.
Mr Cohen, who has not been charged, is reportedly under investigation for possible bank and tax fraud, as well as potential violation of election law.
But he said no money was actually ever transferred, and the recording itself amounted to "powerful exculpatory evidence".
On the tape, Trump can be heard urging Cohen to make sure he properly documents the agreement to buy the rights and urges him to use a check - rather than cash - to keep a record of the transaction, the person said. The source also said that Cohen "is no longer taking a bullet for you (Trump), is no longer a flunky", calling his declaration of independence "his July 4th moment". CNN's Keith Allen has also just reported that Trump's reaction when he was informed that at least one tape was in the FBI's possession, was, "I can't believe Michael would do this with me". She turned down the offer and began a relationship that included, she said, interactions between the two "many dozens of times".
Trump blames Putin personally for election meddling in latest reversal
In a CBS interview with anchor Jeff Glor , Trump said that he held Putin "personally responsible" for the election interference. The President spoke to key lawmakers and outside confidants leading up to his remarks, people familiar with the matter said.
In the lead-up to the 2016 presidential election, Ms McDougal sold her story to the National Enquirer, which is owned by a personal friend of Mr Trump. But the agreement meant that McDougal signed a non-disclosure agreement that prevented her from revealing the affair. In New York, it is completely legal for a lawyer to tape a client.